# Thread: Need help in squared differential EQn.

1. ## Need help in squared differential EQn.

So my problem is this: "A non-negative function f has the property that its ordinate set over an arbitrary interval has an area proportional to the arc length of the graph above the interval. Find f"

So i started with the integral of f and the definition of arc length in 2-space integral ( sqrt(1 + f ' (x)^2)) and then set them equal to each other. but then i get this:

f(x) ^ 2 = 1 + f ' (x) ^ 2

But I don't know how to solve this. Any help will be great.

2. Originally Posted by hashshashin715
So my problem is this: "A non-negative function f has the property that its ordinate set over an arbitrary interval has an area proportional to the arc length of the graph above the interval. Find f"

So i started with the integral of f and the definition of arc length in 2-space integral ( sqrt(1 + f ' (x)^2)) and then set them equal to each other. but then i get this:

f(x) ^ 2 = 1 + f ' (x) ^ 2

But I don't know how to solve this. Any help will be great.
You can do one of two things. First, solve for $\displaystyle f'$ so

$\displaystyle f' = \pm \sqrt{f^2-1}$ which is now separable.

$\displaystyle 2 f f' = 2 f' f'' \; \text{or}\; f'(f''-f) = 0$ which gives two cases $\displaystyle f' = 0$ or $\displaystyle f''-f=0$.

$\displaystyle f = k \sqrt{1 + f'^2}$